Transfer of null and meagre invariants between Cantor space and the line

Let $E\subseteq 2^{\omega }$ be the countable set of eventually constant binary sequences, and let $D\subseteq \mathbb{R}$ be the dyadic rationals. The binary-value map $b:2^{\omega }\setminus E\to (0,1)\setminus D$ is a bijection. It is a homeomorphism: a finite prefix fixes a dyadic interval; at a nondyadic point that interval can be made arbitrarily small, and every sufficiently small neighborhood avoiding adjacent dyadic endpoints fixes a finite prefix. The inverse binary digit map sends a length-$m$ cylinder to the corresponding half-open dyadic interval of length $2^{-m}$ after the countable dyadic exceptions are removed. Thus [F1] and the $\pi$-$\lambda$ argument show that binary value pushes fair-coin measure to Lebesgue measure on $(0,1)$; the discarded countable sets have measure zero. Therefore $b$ and its inverse preserve null sets: arbitrary null subsets are contained in Borel null hulls; restricting to the two conull cores preserves the completed null ideals. A homeomorphism preserves meagreness on the cores.

The complement $\mathbb{R}\setminus D$ splits into the disjoint relative clopen components $I_z=(z,z+1)\setminus D$ for $z\in \mathbb{Z}$. The Cantor core $2^{\omega }\setminus E$ splits into the disjoint relative clopen components $J_k=[0^{k}1]\setminus E$ for $k\in \omega$. Choose a fixed bijection $z\leftrightarrow k$. On $I_z$ first translate by $-z$, then apply $b^{-1}$, then prefix the resulting binary sequence by $0^{k}1$. This gives a homeomorphism $h$ between the two cores. On this component, prefixing scales fair-coin measure by the positive constant $2^{-(k+1)}$: this holds first for cylinders by [F1], then for Borel sets by the $\pi$-$\lambda$ argument and for arbitrary null subsets by Borel hulls. Translation preserves Lebesgue null sets. Thus $h$ preserves null sets in both directions, as well as relative meagre sets on the two cores. Countable unions across the components preserve both ideal properties.

Both omitted sets $D$ and $E$ are countably infinite, so both cores are dense in their ambient spaces. For any dense subspace $Y\subseteq X$ and $A\subseteq Y$, $A$ is nowhere dense in $Y$ exactly when it is nowhere dense in $X$: if ${\bar{A}}^X$ contained a nonempty open $U$, then $U\cap Y$ would be a nonempty relative open subset of ${\bar{A}}^Y$; conversely, if ${\bar{A}}^Y$ contained a nonempty relative open $U\cap Y$, density of $Y$ would force $U\subseteq {\bar{A}}^X$. Taking countable unions gives the same equivalence for meagreness. Extend $h$ by a fixed bijection $D\to E$ to a bijection $H:\mathbb{R}\to 2^{\omega }$. For any $A\subseteq \mathbb{R}$, the difference between $H[A]$ and $h[A\setminus D]$ is a subset of $E$. Conversely, the difference between $H^{-1}[B]$ and $h^{-1}[B\setminus E]$ is a subset of $D$. By [F3], both null and meagre ideal membership are therefore preserved in both directions by $H$.

The bijection $H$ preserves ideal membership, unions, and set containment. Since [F2] gives the real-line minima, $H$ induces bijections between the candidate-witness collections in the four definitions and their Cantor-space counterparts, so the Cantor minima also exist and have the same values. Applying $H^{-1}$ gives the reverse cardinal inequalities and hence equality for all eight invariants. AC is used to compare cardinalities of arbitrary witness families in [F2]; the component and exceptional-set matchings themselves are explicit. ∎